Gate-tunable coherent perfect absorption of terahertz radiation in graphene

Perfect absorption of radiation in a graphene sheet may play a pivotal role in the realization of technologically relevant optoelectronic devices. In particular, perfect absorption of radiation in the terahertz (THz) spectral range would tremendously boost the utility of graphene in this difficult range of photon energies, which still lacks cheap and robust devices operating at room temperature. In this work we show that unpatterned graphene flakes deposited on appropriate substrates can display gate-tunable coherent perfect absorption (CPA) in the THz spectral range. We present theoretical estimates for the CPA operating frequency as a function of doping, which take into account the presence of common sources of disorder in graphene samples.

As we shall see, it is possible to achieve coherent perfect absorption (CPA) at a specific frequency in an unpatterned graphene sheet, provided the graphene is doped to a specific level. CPA is a manifestation of critical coupling, a phenomenon that is commonly exploited in integrated photonics [11,12]. In critically-coupled systems, the multiple coherent paths taken by incident photons to exit an absorbing system-by scattering or reflection-interfere destructively, resulting in 100% conversion into the absorption channel. (The CPA phenomenon can be studied in a general optical scattering context [13,14], but here we consider a single input and output channel. Single-channel critical coupling, utilizing a thin absorbing layer of organic aggregate, has previously been demonstrated experimentally [15].) In the context of unpatterned graphene, CPA requires a specific universal value of e(σ), the real part of the optical sheet conductivity. Using random phase approximation (RPA)-Boltzmann transport theory [3, [16][17][18], we derive the frequency at which this condition is met, for each doping level. We demonstrate the existence of a threshold (minimum) doping level which depends on the impurity density and is typically around 200 meV. The operating frequency for CPA lies in the THz range, up to ∼ 4 THz for realistic doping levels and substrates. The only additional requirements for CPA to occur are that the substrate should be perfectly reflective and that the phase of its reflection coefficient should have a specific value; in particular, the optical quality factor does not enter into the result. CPA of THz radiation in graphene may yield a significant improvement of the current performance of graphene-based THz photodetectors [19].
Previous works have suggested enhancing optical absorption in graphene by placing it in an optical cavity. At the frequency of a cavity resonance, the total absorption, due to multiple passes of recycled photons, can be much larger than the single-pass absorption, in principle reaching 100%. Perfect absorption has recently been demonstrated experimentally with a Fabry-Pérot arXiv:1402.2368v1 [physics.optics] 11 Feb 2014 microcavity [20,21], and enhanced absorption of ∼ 80% has been demonstrated in an attenuated total reflectance configuration [22]. For applications in the far-infrared and THz, another promising route towards enhanced or even perfect absorption is to pattern the graphene, in order to exploit graphene's extremely well confined plasmons [8,9,23,24]. Thongrattanasiri et al. [25], for example, have argued that 100% light absorption can take place in periodic arrays of doped graphene nanodisks on a substrate that is reflective (via total internal reflection or a metal back-gate). Similar results have been obtained by Nikitin et al. [26] for arrays of graphene ribbons.
Unlike the configurations where graphene is embedded in an optical cavity, we consider a graphene sheet which is connected directly to free space, which limits photon recycling. At odds with Refs. 25, 26, we furthermore focus on unpatterned graphene sheets. Liu et al. have theoretically studied the enhancement of absorption of graphene in a similar configuration and have argued that ∼ 80% absorption at 1.2 THz is possible with a substrate containing a metal back reflector [27,28], which they attribute to photon localization in the graphene sheet [27]. In this paper, we argue that perfect (100%) absorption by single-layer graphene can be achieved, with critical coupling as the operating principle. Our results might be generalized to other materials with conducting surface states, such as topological insulators [29].
Theory of CPA in a doped graphene sheet.-Consider a 2D conducting sheet suspended in air, parallel to thê x-ŷ plane. For normally-incident electromagnetic plane waves propagating in theẑ direction the transfer matrix across the sheet is where 1 1 is the 2 × 2 identity matrix and R ≡ σ/(2σ a ) with σ the optical sheet conductivity and Note that σ a = σ uni /(πα) ∼ 44 σ uni . Eq. (1) implies the well-known fact that such a sheet can absorb at most 50% of light incident from one side. This limit can be overcome by placing an optical cavity (or "substrate") on the opposite side of the sheet. Let us assume that the substrate is perfectly reflecting, with a complex reflection coefficient e iφ . Then the reflection coefficient r for the entire system (sheet plus substrate) satisfies where d is the wave intensity on the mirror side of the graphene sheet. Hence, from Eq. (1), The absorbance is then given by A = 1 − |r| 2 . Note that material losses in the substrate can be modeled by setting m[φ] > 0; however, we will assume no such losses. From Eq. (4), we see that CPA, i.e. r = 0, occurs when the conductivity σ ≡ e(σ) + i m(σ) satisfies Note that as long as e(σ) reaches the value σ a , then regardless of the value of m(σ) there is always some value of φ that satisfies the second part of Eq. (5). Eq. (5) also implies that CPA cannot occur in an ideal undoped graphene sheet since, as we noticed earlier after Eq.
(2), σ uni ∼ σ a /44 σ a . In a real doped graphene sheet, however, the optical conductivity can be much larger than σ uni at frequencies ω 2ε F / ≡ 2ω F where ε F is the Fermi energy. In this regime, the inter-band contribution to the conductivity is negligible and σ as a function of frequency ω is well described by the Drude formula [30][31][32][33]: where σ dc = 2(e 2 /h)ω F τ is the dc conductivity and τ is the dc transport scattering time. Comparing to Eq. (5), we find that CPA occurs at the frequency For this to be satisfied, we require .
The dc transport scattering time τ depends on the doping level in a way that is controlled by the various electronimpurity and electron-phonon scattering mechanisms. In this work we concentrate on the former, which dominates at sufficiently low temperatures, and, more precisely, we consider scattering of electrons against charged and short-range impurities. We denote by the symbol τ l (τ s ) the contribution to τ that stems from charged impurity (short-range disorder) scattering: In the regime of doping we are interested in, both contributions can be safely determined from Boltzmann transport theory (in conjunction with the RPA to treat screening) [3, [16][17][18]. For simplicity, we assume that the charged impurities are located on the graphene sheet: this assumption reduces the number of parameters of the theory, allows a complete analytical treatment of charged-impurity scattering, and can be easily relaxed by allowing a finite average distance d between the impurities and graphene. In the case d = 0 the result is where with u(x) = arccos(1/x)/ √ x 2 − 1. Here, n is the carrier density, related to the Fermi frequency by ω F = v F k F where v F is the density-independent Fermi velocity (∼ 10 6 m/s) and k F = √ πn is the Fermi wave number; n i is the density of charged impurities; and α ee = e 2 /( v F ), with the average dielectric constant of the materials surrounding the graphene flake, is the so-called "graphene fine structure constant" [34], i.e. the ratio of the electronelectron interaction energy scale (e 2 k F / ) to the kinetic energy scale ( v F k F ). In Eq. (11), σ s is the limiting conductivity when the scattering is purely short-range. The value of σ s can be determined [3] by fitting the sub-linear dependence of the conductivity on density in the highdensity regime.
Applying Eqs. (9)-(12) to Eq. (8), we find that CPA requires The first inequality in (13) sets the limits for the number of defects in graphene (such as vacancies or chemisorbed dopants) and is easily satisfied in most experiments on exfoliated and CVD grown graphene without the intentional addition of defects, where σ s 200 e 2 /h ∼ 3σ a . The second inequality describes the minimum carrier doping that must be present (relative to the charged impurity density). For a relatively large impurity concentration of n i = 10 12 cm −2 , this corresponds to a minimum doping of 270 meV, which can also be attained in most graphene experiments [3, 17,35]. Therefore, the CPA criteria can be easily achieved using graphene samples of comparable quality to what is routinely made in the experimental literature.
Numerical results and discussion.- Fig. 1 shows the operating frequency for CPA, ω a , as a function of ε F , as calculated from Eq. (7) with Eqs. (9)- (12). We observe a rapid increase in ω a with ε F above the threshold doping level, which, as noted above, depends on n i and σ s . When short-range scattering is negligible (σ s σ a ), ω a saturates for large dopings at the value For n i = 10 12 cm −2 and α ee = 0.8, this corresponds to a maximum frequency of ω (0) a /(2π) ≈ 1.7 THz. When σ s is non-negligible, ω a does not saturate with increasing ε F , but instead increases linearly at high doping levels. These operating frequencies correspond to energies in the 10 meV range (3 THz ∼ 12.4 meV), much lower than the doping level, which is ∼ 200 meV. Hence, the Drude conductivity model in Eq. (6) is valid throughout. When the system is tuned for perfect absorption at ω a , light incident at nearby frequencies is also strongly absorbed. The absorption frequency bandwidth is determined by τ as well as the frequency dispersion of the substrate reflection phase φ in Eq. (5). The τ -limited frequency bandwidth is extremely large. In Eq. (4), assuming that φ is frequency-independent, if perfect absorption occurs at frequency ω a , then at frequency ω, Eq. (6) then gives, for the absorbance, Hence, the absorption bandwidth would simply be the inverse of the transport scattering time. For the parameters we have considered, τ −1 ∼ 10 THz, on the order of ω a itself. For practical purposes, therefore, the absorption bandwidth is limited by the detuning of φ(ω). The substrate could be engineered to minimize this detuning; for example, one could use low-order modes for which φ has bandwidth on the order of the operating frequency. Fig. 2 shows the absorbance A(ω) for a graphene sheet on a SiO 2 substrate with a metal reflector on the far end, which is tuned to produce perfect absorption at ω a /(2π) = 1.4 THz. The system operates at the lowestorder cavity mode, with the cavity length being approximately equal to a quarter-wavelength. A broad absorption resonance, of relative bandwidth 0.1, is indeed observed. The bandwidth could be further optimized by designing a non-uniform cavity. By comparison, in the experiments demonstrating the principle of CPA in optical cavities, the relative absorption bandwidth was ∼ 10 −3 [13,14].
The absorption is robust against variations in the incidence angle. For oblique incidence, two different polarizations should be considered: transverse magnetic (TM; magnetic field parallel to the plane of the graphene sheet) and transverse electric (TE; electric field parallel to the plane). For incidence angle θ, the transfer matrix across the graphene sheet is The reflection coefficients are thus obtained by replacing R with R(cos θ) ±1 in Eq. (4). The cavity phase delay φ also depends on θ. Supposing we have optimized the structure for perfect absorption at normal incidence [36] (θ = 0), the reflectance can be calculated as a function of θ for each polarization; the result is This is demonstrated numerically in Fig. 3(a). The system can also be detuned from the CPA condition by varying the doping level, e.g. via the electrochemical potential of the metallic gate acting as the backreflector [25]. As shown in Fig. 3(b), one can tune between perfect absorption at the tailored doping level and close to zero absorption at low doping levels.
As a final note, we would like to point out that even if the conductivity is nowhere near the magic value σ a , the total absorption can generally be reduced to zero. This suppression of absorption is an interference effect. The absorptivity of the graphene sheet remains constant, but the total absorption goes to zero since the local intensity at the graphene sheet vanishes (i.e. the sheet lies at a node of a standing wave). This effect is similar to those reported in Ref. 37, where the absorption goes to zero (a large value) when there is a node (anti-node) at the position of the thin absorbing layer.
Acknowledgements.-CYD acknowledges support from the National Research Foundation Singapore under its